https://doi.org/10.1140/epjc/s10052-019-7198-1 Regular Article - Theoretical Physics
Constraints on interacting dynamical dark energy and a new test
for
CDM
Marco Bonici1,2,a, Nicola Maggiore1,2,b
1Dipartimento di Fisica, Università di Genova, via Dodecaneso 33, 16146 Genova, Italy 2I.N.F.N.-Sezione di Genova, via Dodecaneso 33, 16146 Genova, Italy
Received: 25 March 2019 / Accepted: 4 August 2019 © The Author(s) 2019
Abstract We consider a generic description of interact-ing dynamical Dark Energy, characterized by an equation of state with a time dependent coefficientw(t), and which may interact with both radiation and matter. Without referring to any particular cosmological model, we find a differential equation which must be satisfied byw(t) and involving the function Q(t) which describes the interaction between Dark Energy and the other cosmological fluids. The relation we find represents a constraint for various models of interact-ing dynamical Dark Energy. In addition, an observable is proposed, depending on kinematic variables and on density parameters, which may serve as a new test forCDM.
1 Introduction
The accelerating expansion of our Universe [1,2] can be described by a cosmological constant in the Einstein-Hilbert action of General Relativity, introduced by Einstein [3] in order to have a static Universe. There are a few well known good reasons to be unsatisfied with the description of Dark Energy (DE) in terms of a cosmological constant, but the known difficulties, or seemingly unnatural coincidences, can be solved invoking very peculiar initial conditions [4]. In the hopeful wait of an experimental conclusive evidence, theo-rists since long time provided us with a variety of alternative models for DE [5], with the request that any cosmological model should reproduce an Universe which, at our epoch, is almost perfectly flat and filled by matter and DE in the ratio of about 3/7, where the DE is effectively approximated by a constant. In Literature many “dynamical” alternatives for DE can be found, like, for instance, the quintessence models [6,7], where the role of the cosmological constant is played by scalar potentials, suitably parametrized to get the desired
ae-mail:marco.bonici@ge.infn.it be-mail:nicola.maggiore@ge.infn.it
behavior, and the K-essence models [8–10], likewise built in terms of scalar fields, where the accelerated expansion of the Universe is driven by the kinetic term. Both quintessence and K-essence models belong to the wider category of modified theories of gravity, whose purpose is to extend their range of validity to large, galactic, scales. In the most general case, any dynamical, as opposed to constant, model for DE may interact with all the components of the cosmological perfect fluid in terms of which is written the energy momentum ten-sor appearing at the right hand side of the Einstein equations Rμν−1
2gμνR= 8πG Tμν. (1.1)
The coupling could be minimal, through the metric depen-dent invariant measure, or minimal, with direct and non-trivial coupling with gravity, like in the scalar-tensor theo-ries, or by means of direct interactions with baryonic matter, and/or with neutrinos, and/or with Dark Matter [11–16].
In this paper we keep a very general perspective. Without referring to any particular DE model, we assume only that DE is realized by means of a perfect fluid satisfying an Equation of State (EoS) with a time dependentw-coefficient
pDE= w(t)ρDE (1.2)
and that DE interacts non-minimally with any cosmological component. The interactions result in broken covariant con-servation laws of the energy momentum tensors of the single cosmological components
∇μ(Ti)μν = (Qi)ν i = matter, radiation, DE (1.3) keeping the total energy momentum tensor conserved. The aim of this paper is to give a criterion to select amongst different models of interacting dynamical DE, assuming only the validity of the Friedmann equations for the scale factor appearing in the Robertson–Walker metric. This subject has been faced following different strategies [17–26], all of which need some kind of assumptions, on the phenomenological
form of the interactions Qi, or on the choice of the potential in the quintessence models, for instance. In this paper, we try to be as much general as possible, adopting a model independent cosmographic approach (see [27] for an updated review).
In order to reach this goal, in Sect.2we relate the time derivative of the DE EoS coefficientw(t) to the interactions Q(t), by means of the kinematic variables associated to the scale factor a(t): the Hubble parameter H(t), the deceler-ation q(t) and the jerk j(t), sometimes called statefinder r(t)-parameter [28]. In Sect.3 we discuss the implications of our analysis for theCDM model and we propose an observable, written in terms of kinematic variables and den-sity parameters, whose non-vanishing value would imply a failure ofCDM. In the concluding Sect.4we summarize and discuss our results.
2 Constraints on interacting dynamical dark energy The energy momentum tensor for a cosmological perfect fluid is:
Tμν= (ρ + p)UμUν+ pgμν, (2.1)
where Uμis the fluid four-velocity,ρ is the rest-frame energy density and p is the isotropic rest-frame pressure. The EoS relates pressure and energy density and its general form is:
p= p(ρ), (2.2)
whose simplest case is represented by the linear relation
p= wρ, (2.3)
wherew is a coefficient not depending from the energy den-sityρ.
Following [29], we consider here the more general EoS (2.2), whose Taylor expansion around the energy density at the present epochρ0= ρ(t)|t=t0 is [29]
p(ρ) = p0+ κ0(ρ − ρ0) + O[(ρ − ρ0)2], (2.4) where p0= p(ρ0) and κ0= d pdρ
t=t0 .1
The aim is to express the first two coefficients of the above expansion in terms of the scale factor a(t) appearing in the Robertson - Walker metric
ds2= −dt2+ a2(t) dr2 1− kr2+ r 2(dθ2+ sin2θdφ2) , (2.5) where k is a constant parameter related to the spatial cur-vature: k = 0, k > 0 and k < 0 for flat, closed and open Universes, respectively.
1From now on,O
0 ≡ O(t)|t=t0, for any observableO(t), where t0
stands for the present epoch.
More precisely, we would like to write p0andκ0in terms of the kinematic variables related to the dimensionless time derivatives of a(t), namely the Hubble parameter H(t), the deceleration q(t) and the jerk j(t), which are observables quantities, thus making this approach independent from a particular cosmological model. The kinematic variables are defined as follows H = ˙a a (2.6) q = − ¨a a H2 (2.7) j= ... a a H3. (2.8)
It is customary to suppose that the cosmological fluid is an incoherent mixture of the three forms of canonical fluids (i = 1 matter, i = 2 radiation, i = 3 DE represented by a cosmological constant) plus, following a standard nota-tion [30], the spatial curvature contribution (i = 4 ), each satisfying the linear EoS
pi = wiρi. (2.9)
Consequently, the EoS (2.3) takes the form 4 i=1 pi = 4 i=1 wiρi. (2.10)
Once we define the density parameters i =
8πG
3H2ρi, (2.11)
the Friedmann equation can be written
i
i = 1. (2.12)
As done in [29], we denote with O the average of generic physical observablesOiweighted by the density parameters i of each fluid [29]: O ≡ 4 i Oi i. (2.13)
Consequently, from the EoS (2.3) we have w = p ρ = i pi iρi = iwiρi iρi = iwi i i i = i wi i = w. (2.14)
In this paper we introduce the following two generaliza-tions with respect to the approach described in [29]:
1. We allow a time dependence of the EoS coefficientswi appearing in (2.9)
Even though we are mostly interested in physical situa-tions where only the DE fluid may have a time dependent w3(t), for the moment we take a more general attitude. The known scalar quintessence model for DE is an exam-ple of DE fluid with a time dependent EoS coefficient, but we point up that in this paper we do not necessarily limit ourselves to this particular case.
2. The energy momentum tensor (2.1) is given by the sum of the different components of the perfect cosmological fluid. We give the possibility to each component to break the conservation law:
∇μ(Ti)μν = (Qi)ν, (2.16) keeping the total energy momentum tensor conserved, which implies a constraint on the breakings
∇μTμν = 0 ⇒ i
(Qi)ν = 0. (2.17)
In most cases, only the matter and DE components of the energy momentum tensor possibly display a break-ing of the conservation law in the late Universe, not the radiation nor the curvature contributions. Again, for the moment we stay on general grounds, and the breakings Qi, which, because of the constraint (2.17) must be at least two, physically correspond to interactions between the cosmological components fluids. Examples of non-vanishing DE interactions are given in [17–24].
Deriving both sides of the Friedmann equation (2.12) with respect to time, we have
i ˙ i = 0 ⇒ i d dt ρi H2 = 0. (2.18)
To calculate ˙ρi, we use the covariant conservation of the energy momentum tensor (2.1), The ν = 0 component of (2.16) gives
˙ρi = −3H(ρi+ pi) + Qi, (2.19)
where we defined
Qi ≡ −(Qi)0= +(Qi)0. (2.20)
On the other hand
˙ H= ¨a a − ˙a a 2 = −H2(1 + q), (2.21) where we used the definition (2.7) of the deceleration q(t).
Coming back to Eq. (2.18), we can now write
0= i ˙ρ i H2− 2 H3ρiH˙ (2.22) = i −3H(1 + w i)ρi H2 + Qi H2 + 2 H3ρiH 2(1 + q) (2.23) = −3H(1 + w) + 2H(1 + q), (2.24)
where we used (2.19), (2.9) and (2.21), and we used the definition of weighted average (2.13) for the quantitieswi(t) and the constraint (2.17) on the breakings Qi.
Therefore, using (2.14), the following relation holds w = w = 2q− 1
3 , (2.25)
which, in particular, relates the first coefficient of the Taylor expansion of the EoS (2.4) to the deceleration q(t), since
p0= w0ρ0=
2q0− 1
3 ρ0. (2.26)
Notice that the relation (2.25), which has been derived in [29] for non-interacting DE with constant EoSw-parameter, has a general validity, since it holds also for ˙wi = 0 and Qi = 0. Let us now considerκ0, the second coefficient of the EoS Taylor expansion (2.4): κ = d p dρ = ˙p ˙ρ = i ˙pi i ˙ρi = i(wi˙ρi+ ˙wiρi) i ˙ρi = i{wi[−3H(1 + wi)ρi+ Qi] + ˙wiρi} i[(−3H)(1 + wi)ρi+ Qi] = i[(−3H)(wi+ wi2)ρi+ wiQi+ ˙wiρi] i(−3H)(1 + wi)ρi = w + w2 1+ w − 8πG 9H3 iwiQi 1+ w − ˙w 3H(1 + w), (2.27) where we took into account (2.19), (2.17) and (2.13). We need ˙w, i.e. the weighted average of the time derivatives of the EoS coefficients wi(t), which vanish in the CDM model. To obtain it, we look for an expression for the time derivative of the weighted average ˙w:
˙w = i d dt(wi i) = ˙w + 8πG 3 i wi d dt ρi H2 = ˙w +8πG 3 i ˙ρi H2 − ρi 2 H3H˙ = ˙w+8πG 3 i (−3H)(1+wi)ρi+Qi H2 +ρi 2 H(1+q) = ˙w−3H(w + w2)+8πG 3H2 i wiQi+3Hw(1 + w), (2.28) where, in the last row, we used (2.25) to eliminate the decel-eration q in favor ofw. Introducing the variance of the values wi
σ2
we get ˙w = ˙w + 3Hσ2 w− 8πG 3H2 i wiQi. (2.30)
It is easily seen that, using in (2.27) the above expression (2.30) for ˙w and the definitions of the deceleration q(t) (2.7) and of the jerk j(t) (2.8), we finally get
κ = d pdρ = j− 1
3(1 + q), (2.31)
which, asw(t) (2.25), is an universal quantity, whose expres-sion is valid whether ˙wi = 0 and Qi = 0 or not.
Let us take for a moment Eq. (2.30) at ˙wi = Qi = 0, which is the standard case we are generalizing in this paper. The variance (2.29) reduces to
σ2
w= −3H˙w = 29[ j − q(1 + 2q)], (2.32)
where w(t) in (2.25) and the definition (2.8) of the jerk j(t) have been used. The above expression for σw2 tells us how the weighted accuracy on the estimate of thewi, assumed to be constant, evolves in time, driven by the time dependence of the cosmological parameters ionly. In gen-eral, it is not allowed to deduce that the right hand side of (2.32) is non-negative, since the weights i present in σ2
w = i(wi− w)2 i may be negative. Indeed, while 1 and 2are certainly non-negative functions of time, since they are related to matter and radiation energy density respec-tively, the density parameters 3and 4, which refer to DE and curvature, might, in principle, have any sign. What we can state, is that, at our epoch, (0)1 0.3, (0)2 0 and (0)3 0.7 [1], and, consequently, that the Universe, in excel-lent approximation, is spatially flat k 0. Therefore, at our epoch, but not at any time, the right hand side of (2.32) is non-negative
j0≥ q0(1 + 2q0). (2.33)
Equation (2.33) is a constraint which must be satisfied, at t= t0, by the kinematic variables related to the time derivatives of the scale factor a0, namely the deceleration q0and the jerk
j0.
In case of non-vanishing ˙wi and Qi, the more general relation (2.30) represents a constraint for interacting dynam-ical DE. In fact, it relates the possible time dependentwi(t) appearing in the EoS (2.9) of the cosmological fluids (2.3) to their corresponding interactions (2.16). At the present epoch t= t0we have: ˙w+3Hσ2 wt=t 0 ≡ K0= i ˙wi i+ 8πG 3H2wiQi t=t0 , (2.34)
where K0is a physical observable, depending on measurable quantities (density parameters and kinematic variables).
Making the reasonable assumption that only the DE com-ponent of the cosmological perfect fluid may have an EoS of the form (2.9) with ˙w3(t) = 0, and observing that the rele-vant interactions are the ones involving DE, which translates into Q3= 0, the relation (2.34) at t = t0reduces to
˙w3 3+ 8πG 3H2w3Q3 t=t0 = K0, (2.35)
where we used the fact that, for matter,w1strictly vanishes. Since the values at our epoch of the DE density parameter (0)3 , of the coefficient of the DE EoSw
(0)
3 , of the Hubble constant H0 and of the quantity K0, are known, the rela-tion (2.35) represents a constraint on the possible theoretical models of interacting dynamical DE, in particular on the time dependence of the DE EoS coefficient ˙w3(0) and on the DE interaction Q(0)3
w3(t) = w(0)3 + ˙w3(0)(t − t0) + O(t2) (2.36)
Q3(t) = Q(0)3 + O(t). (2.37)
It is a remarkable and, to our knowledge, so far unknown fact that the interactions involving DE and its dynamical EoS are not independent one from each other.
3 A new test forCDM model
The aim of this paper is to put constraints, mainly by means of the relation (2.35), on the possible models of DE, with particular concern on the DE EoS and on the interaction DE-matter. In order to be able to make a comparison, it is useful to summarize what is predicted by the Standard Model of Cosmology.
After the observational evidence from supernovae for an accelerating Universe and a cosmological constant [1], we know that, at our epoch, our Universe is filled by DE and (mostly dark) matter :
(0)1 0.3 ; (0)3 0.7, (3.1)
where we used the notations adopted in this paper, according to which the subscripts 1 and 3 stand for matter and DE, respectively. An immediate consequence of the Friedmann equation, is that our Universe is almost flat
k 0, (3.2)
since, at our epoch, the radiation contribution to the whole cosmological perfect fluid being is highly suppressed:
(0)2 0. (3.3)
TheCDM model well describes this scenario, where the DE is realized by means of a cosmological constant. The
Table 1 EoS coefficients and density parameters inCDM
i = 1: matter i = 2: radiation i = 3: DE i = 4: curvature
(0)i 0.3 0 0.7 0
wi 0 1/3 − 1 − 1/3
CDM situation, including the EoS coefficients wi of the single cosmological fluids, is summarized in the following Table1.
In theCDM model the only EoS coefficient which sur-vives isw3. Its value (w3= −1) corresponds to the contribu-tion to the cosmological fluid coming from the cosmological constant.
According to theCDM model, the jerk variable (2.8) should be constant, and in particular
CDM ⇒ j(t) = 1. (3.4)
This can be seen in many ways. In the particular framework of this paper, let us consider the expression (2.32) for the variance of the EoS coefficientw, which holds for ˙wi = Qi = 0 and hence true in the CDM model:
j =9 2σ
2
w+ q(2q + 1). (3.5)
From the definition of weighted average (2.13), of variance (2.29) and using the fact that in theCDM model the only non-vanishing EoS coefficient is the DE one, we have σ2
w= w23 3(1 − 3). (3.6)
On the other hand, fromw in (2.25), we get the following relation for the deceleration parameter q(t):
q = 3
2w3 3+ 1
2. (3.7)
Using (3.6) and (3.7), we get the following relation for the jerk parameter
j = 1 +9
2w3(1 + w3) 3, (3.8)
which is equal to one if the DE is described by a cosmological constant, i.e. w3 = −1. Therefore, the jerk parameter is a model independent, kinematic observable valuable to test the CDM model, since any deviation from j = 1 is a signal of alternative descriptions.
Let us now consider K0defined by the left hand side of (2.34)
K0≡ ˙w + 3Hσw2 t=t0
. (3.9)
According to theCDM model, the DE has a constant EoS w-coefficient, and does not interact, hence, from the right hand side of (2.34), K0should vanish identically:
CDM ⇒ K0= 0. (3.10)
From its definition (3.9), it is easy to check that K0can be written in terms of measurable quantities as follows:
K0 H0
= 3 1(1 − 1) − 2
3[ j0− q0(1 + 2q0)] . (3.11) We point out that, analogously to the case concerning the jerk parameter j(t) = 1, a non-vanishing value for K0would be a certain signal of the failure of theCDM model, and we remark that the case K0= 0 is independent of j0= 1. It is easily seen, in fact, that observational situations are possible where j0 = 1 and K0 = 0 at the same time, for which we should conclude againstCDM.
On the other hand, even a K0compatible with zero would not represent a confirmation ofCDM. Both cases K0 = 0 and K0 = 0, in fact, could be realized by means of an interacting dynamical DE, with ˙w3 = 0 and/or Q3 = 0. Once again, our point of view is to test and constrain possible models of interacting dynamical DE, assuming thatCDM is a model which well describes, “only” in an effective way, the observations on the Universe so far.
The observable K0is written in terms of measurable vari-ables, through (3.11). A precise estimate of K0 is highly nontrivial, and goes beyond the scope of our paper. There are in fact two kind of difficulties in evaluating K0. The first is that, at the moment, the quantities in terms of which K0is expressed (the Hubble constant H0, the matter EoS parameter 1, and the kinematic variables q(t) and j(t), all evaluated at the present epoch) are known with large errors, in partic-ular the jerk j0, not to mention the known existing tension on the value of H0. The situation will improve drastically in the next future, since, for instance, one of the aims of the forthcoming Euclid experiment is to refine the measure of the kinematic variables, which therefore will be known with much greater accuracy. Under this respect, K0is a variable which we believe will become very interesting in the future. The other difficulty concerns the type of analysis which should be performed. In fact, particular care should be payed in Cosmology when dealing with “experimental” data, which should be treated according to the Bayesian analysis. We are not experts in this kind of analysis, and, even if the observ-ables in terms of which K0 is expressed were known with smaller errors, we prefer to leave this task to professionals, whenever the kinematical variables will be known with more precision.
Therefore, although a precise Bayesian analysis to deter-mine K0 is premature, and although our paper focuses on the formal aspects of a theory of interacting dynamical Dark Energy, in the final part of this section we give a preliminary, albeit rough, estimate of K0based on the publicly available data sets.
Table 2 Deceleration and jerk a b c d
q0 − 0.644 ± 0.223 − 0.6401 ± 0.187 − 0.930 ± 0.218 − 1.2037 ± 0.175
j0 1.961 ± 0.926 1.946 ± 0.871 3.369 ± 1.270 5.423 ± 1.497
Table 3 Estimates of K0 a b c d
K0 − 0.516 ± 0.685 − 0.512 ± 0.622 − 1.026 ± 0.931 − 1.817 ± 1.091
Concerning the deceleration q0and the jerk j0, evaluated at our epoch, we report in Table2four maximum likelihood values, with their 68% confidence intervals:
The observational constraints for the deceleration param-eter q0 and the jerk j0 reported in Table 2 were recently obtained in [31], and a, b, c and d refer to the following different combinations of low redshift datasets:
a: BAO +Masers+TDSL+Pantheon,
where BAO stands for the observations from Baryon-Acoustic-Oscillations [32–36], Masers is the Megamaser Cosmology Project [37–40], TDSL means time-delay in strong lensing measurements by H0LiCOW experiment [41] and Pantheon are the data for SNIa in terms of E(z) [42,43]
b: a + H0measurement done in [44,45]
c: a + H(z) measurements (Hubble parameter data (OHD) as a function of redshift [46])
d: all the data (a + H0+ H(z)).
According to the d-dataset in Table2, which contains all the others, it is apparent that theCDM value j0 = 1 is incompatible with data, at 3.06σ confidence limit.
Concerning K0, it is convenient to consider the dimension-less quantity K0/H0, in order to get rid of the well known tension existing on the Hubble constant [44,45,47,48]. A rough and preliminary estimate, which takes into account the values of q0and j0given above and the value of the mat-ter density paramemat-ter 1, which, according to the latest SN Ia measurements from the Pantheon Catalogue [49], is
1= 0.298 ± 0.022, (3.12)
gives the four values listed in Table3.
All the above values of K0are compatible with theCDM value K0= 0 within 1 to 2σ. Therefore, according to the data available so far, there is no evidence againstCDM model. A more accurate, constrained analysis might be done following the Bayesian methods in cosmology [50], but for the moment our aim is just to give an estimate of the right hand side of our result (2.35), by means of observable quantities.
4 Conclusions
The points where the CDM model creaks are more and more. An example of these weaknesses is the well known tension on the measurements of the Hubble constants H0. The value given by the Planck collaboration [47,48] in the framework of theCDM model is incompatible with other, model independent, estimates [44,45]. The inconsistencies become milder if a dynamical DE is invoked [51]. There-fore, there are strong motivations to investigate models of dynamical DE which, in the most general case, displays an EoS with a time dependent coefficientwDE(t) and which may
interact, in principle, with matter and/or radiation through a (partial) breaking QDE(t) of the covariant conservation of the
energy momentum tensor.
The main and new result of this paper is represented by the relation (2.34), which must be satisfied, at any time, by any model of interacting dynamical DE
DE˙wDE+
8πG
3H2QDEwDE= K, (4.1)
where K(t) is expressed in terms of measurable quantities. The above equation is a differential equation for the DE EoS parameterwDE(t) with time dependent coefficients, one of
which is the interaction QDE(t). It must hold at any time, in
particular must be satisfied by phantom DE models, crossing theCDM point w = −1 in both directions. The equation (4.1) is a constraint on the possible parametrization of Dark Energy which, in its most general dynamical form, turns out to depend on its interactions, with Dark Matter in particular. It is not surprising that it must be so, but the explicit form of how this mutual dependence is realised was not known so far. We proposed a new observable, which we called K0, which measures the relation between Dark Energy and its interac-tions. The analysis which led to (4.1) is model independent, in the sense that we only assumed that the scale factor a(t) appearing in the Robertson–Walker metric (2.5) obeys the Friedmann equation (2.12) and that the quantities involved in K(t) are the density parameters and the kinematic vari-ables, hence are directly measurable.
An important consequence of (4.1) is that, according to theCDM model, it must hold
CDM ⇒ K0= 0. (4.2)
Any deviation from this value must be interpreted as a fail-ure of theCDM model. Low-redshift data show that, at present time, j0 = 1 [31], which seems to indicate a failure ofCDM. It would be greatly interesting to give an accu-rate estimate of K0, according to the available observational data, but this task goes beyond the scope of this paper, also because the available observational data, especially for what concerns the kinematical variables q(t) (2.7) and j(t) (2.8), are affected by large errors [29,31], which make difficult any decisive claim within 3σ . Hopefully, the Euclid space mis-sion, whose launch date is expected in 2021, will drastically improve the experimental situation. With this caveat, we gave a preliminary and rough estimate of K0, which is com-patible, within 3σ, with zero, hence with the CDM model. But, again, a much more accurate evaluation will be possible in the future.
Finally, the relation (4.1) can be read in several ways, depending whether the DE is interacting or not (QDE = 0
or QDE= 0). It is important to emphasize this point because
previous attempts to get informations on the Dark Sector rely on particular assumptions. Our result may provide a model independent description of the Dark Sector, as well as a con-straint for generic parametrizations of the EoS coefficients wDEand of the interactions QDE(t).
Data Availability Statement This manuscript has no associated data
or the data will not be deposited. [Authors’ comment: This is a purely theoretical article. No datasets were generated. The datasets which have been addressed during the current study are those contained in Ref [32–
46].]
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP3.
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